Linear Independence of Finite Gabor Systems Determined by Behavior at Infinity

TL;DR

This paper proves the HRT conjecture for finite Gabor systems generated by specific classes of functions with particular behaviors at infinity, including rapidly decaying, analytic, and positive functions.

Contribution

It establishes the HRT conjecture for new classes of functions, expanding the understanding of linear independence in finite Gabor systems.

Findings

HRT conjecture holds for functions decaying faster than exponential

HRT conjecture holds for functions analytic with germs in a Hardy field

HRT conjecture verified for certain positive functions

Abstract

We prove that the HRT (Heil, Ramanathan, and Topiwala) conjecture holds for finite Gabor systems generated by square-integrable functions with certain behavior at infinity. These functions include functions ultimately decaying faster than any exponential function, as well as square-integrable functions ultimately analytic and whose germs are in a Hardy field. Two classes of the latter type of functions are the set of square-integrable logarithmico-exponential functions and the set of square-integrable Pfaffian functions. We also prove the HRT conjecture for certain finite Gabor systems generated by positive functions.